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  "body": "\n## Definition\n\nLet $G$ be an [[abelian group|082GHNPJ53]] written additively, and let $A$ be a subset of $G$.\nThe set $A$ is *sum-free* if there are no $x, y, z \\in A$, not necessarily distinct, with $x + y = z$.\n\n## Remarks\n\nEquivalently, $(A + A) \\cap A = \\emptyset$, where $A + A$ is the [[sumset|XNBN31HQWB]].\nThe set of odd integers is sum-free in $\\Z$: if $x$ and $y$ are odd, then $x + y$ is even, so it is not in the set.\nThe set $\\set{1, 2, 4}$ is not sum-free: with $x = y = 2$ one gets $x + y = 4$, and repeated elements are allowed.\nFor a positive integer $N$, the upper half $\\set{n : N/2 \u003c n \\le N}$ of $\\set{1, \\dots, N}$ is sum-free, since any sum of two of its elements exceeds $N$.\nThe set $\\set{0}$ is never sum-free, since $0 + 0 = 0$; so a sum-free subset of a group does not contain $0$.\n",
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